{"id":"n-5a7a22b91157a0bd","revision":1,"first_index":0,"fingerprint":"f6-55490b1580ef6491","name":"obreschkoff_combination of Triangle of coefficients in expansion of (1+3*x)^n.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["2","3"],["2","9","9"],["2","15","36","27"],["2","21","81","135","81"],["2","27","144","378","486","243"],["2","33","225","810","1620","1701","729"],["2","39","324","1485","4050","6561","5832","2187"],["2","45","441","2457","8505","18711","25515","19683","6561"],["2","51","576","3780","15876","44226","81648","96228","65610","19683"],["2","57","729","5508","27216","91854","214326","341172","354294","216513","59049"],["2","63","900","7695","43740","173502","489888","984150","1377810","1279395","708588","177147"],["2","69","1089","10395","66825","304722","1010394","2453814","4330260","5412825","4546773","2302911","531441"],["2","75","1296","13662","98010","505197","1924560","5484996","11691702","18403605","20785248","15943230","7440174","1594323"],["2","81","1521","17550","138996","799227","3440151","11258676","28146690","53478711","75996063","78298974","55269864","23914845","4782969"],["2","87","1764","22113","191646","1216215","5837832","21579129","61922718","137918781","236432196","306287163","290166786","189724437","76527504","14348907"]],"degrees":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],"row_sums":["5","20","80","320","1280","5120","20480","81920","327680","1310720","5242880","20971520","83886080","335544320","1342177280"],"nonn":true,"definitions":[{"position":0,"source":"derived:obreschkoff_combination","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["2","3"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","3"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162722,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, +inf)","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)","region_exact":null,"edges":["e-b2c5a8e7eea225e1"]}],"region_exact":{},"incoming_edges":[{"id":"e-b2c5a8e7eea225e1","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-5a7a22b91157a0bd","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"af4857868c8be0fa7cc3f1b2e65902cbde72f58d478a6eae1df8f2d18f59cc0d","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{}}],"premises":[{"position":0,"node_id":"n-6a5af5e1168bb679","role":"sequence","name":"Triangle of coefficients in expansion of (1+3*x)^n.","uses":{"claim":"real_rooted","from_index":0,"region":{}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-6a5af5e1168bb679","role":"sequence","name":"Triangle of coefficients in expansion of (1+3*x)^n.","uses":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-011e64b62a2f7067","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A168682","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168682","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-676ca811b9e4ca3b","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A167896","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A167896","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6a7675c10d16d8c5","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A168730","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168730","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-6fa33ebe500dad1f","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A168826","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168826","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-c27d644f74f45c66","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A168922","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168922","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-d4af94aa2ebe233a","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A168874","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168874","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e1c2d22c3cadf971","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A003947","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A003947","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-fbc22e211ff1feb9","kind":"oeis_match","node_a_id":"n-5a7a22b91157a0bd","node_b_id":null,"fingerprint":null,"oeis_id":"A168778","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A168778","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A003947","relation":"match_row_sums"},{"oeis_id":"A167896","relation":"match_row_sums"},{"oeis_id":"A168682","relation":"match_row_sums"},{"oeis_id":"A168730","relation":"match_row_sums"},{"oeis_id":"A168778","relation":"match_row_sums"},{"oeis_id":"A168826","relation":"match_row_sums"},{"oeis_id":"A168874","relation":"match_row_sums"},{"oeis_id":"A168922","relation":"match_row_sums"}],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 2,  3;\n  1: 2,  9,   9;\n  2: 2, 15,  36,   27;\n  3: 2, 21,  81,  135,    81;\n  4: 2, 27, 144,  378,   486,   243;\n  5: 2, 33, 225,  810,  1620,  1701,    729;\n  6: 2, 39, 324, 1485,  4050,  6561,   5832,   2187;\n  7: 2, 45, 441, 2457,  8505, 18711,  25515,  19683,   6561;\n  8: 2, 51, 576, 3780, 15876, 44226,  81648,  96228,  65610,  19683;\n  9: 2, 57, 729, 5508, 27216, 91854, 214326, 341172, 354294, 216513, 59049;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+3*x)*P(n-1,x) for n >= 1, with P(0,x) = 2 + 3*x.","mathematica":"P[0, x_] := 2 + 3*x;\nP[n_, x_] := P[n, x] = Expand[(1 + 3*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 2 + 3*x; for(n = 1, N, my(i = n + 1); v[i] = (1 + 3*x)*v[i - 1]); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-6a5af5e1168bb679","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-b2c5a8e7eea225e1"],"text":"A013610 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A013610 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}